To simplify the expression \( \frac{5^{-9}}{4^{-12}} \) using the property of negative integer exponents, we can apply the following property:
\[ a^{-n} = \frac{1}{a^n} \]
First, we rewrite the expression:
\[ \frac{5^{-9}}{4^{-12}} = 5^{-9} \cdot 4^{12} \]
Next, we can express \( 4^{12} \) in terms of \( 2 \):
\[ 4 = 2^2 \quad \Rightarrow \quad 4^{12} = (2^2)^{12} = 2^{24} \]
Thus, the expression becomes:
\[ 5^{-9} \cdot 2^{24} \]
Now we also apply the property of negative exponents to \( 5^{-9} \):
\[ 5^{-9} = \frac{1}{5^9} \]
So we have:
\[ \frac{1}{5^9} \cdot 2^{24} = \frac{2^{24}}{5^9} \]
Now, the final expression, which consists of only positive exponents, is:
\[ \frac{2^{24}}{5^9} \]
Thus,
\[ 5^{-9}/4^{-12} = \frac{2^{24}}{5^9} \]